Complex Number Calculator
Enter z₁ = a + bi and z₂ = c + di, pick an operation, and see the answer in rectangular, polar and exponential form with a worked step and an Argand diagram.
Your numbers
Result: z₁ ÷ z₂
w = −1 + 2i
Polar: 2.2361(cos 116.57° + i sin 116.57°)
Exponential: 2.2361·e^(2.0344i)
How it works
Multiply top and bottom by the conjugate of z₂:
[(ac + bd) + (bc − ad)i] / (c² + d²)
= [-5 + 10i] / 5
Argand diagram
z₁ = 3 + 4i
Polar: 5(cos 53.13° + i sin 53.13°)
Exponential: 5·e^(0.9273i)
z₂ = 1 − 2i
Polar: 2.2361(cos −63.43° + i sin −63.43°)
Exponential: 2.2361·e^(−1.1071i)
A complex number a + bi has a real part a and an imaginary part b, where i² = −1. On the Argand diagram it is the point (a, b), and its modulus |z| = √(a² + b²) is the distance from the origin. The argument θ is the angle from the positive real axis, measured here between −180° and 180°.
Addition and subtraction combine the real and imaginary parts separately. Multiplication expands the brackets and uses i² = −1. Division multiplies the top and bottom by the conjugate of the divisor, which turns the denominator into the real number c² + d².
Powers and roots are easiest in polar form. De Moivre's theorem says (r(cos θ + i sin θ))ⁿ = rⁿ(cos nθ + i sin nθ), and every non-zero complex number has exactly two square roots, which point in opposite directions on the Argand diagram.
Frequently asked questions
How do you divide complex numbers?
Multiply the numerator and denominator by the conjugate of the denominator. For (a + bi) ÷ (c + di) this gives [(ac + bd) + (bc − ad)i] ÷ (c² + d²). For example, (3 + 4i) ÷ (1 − 2i) = −1 + 2i.
What is the polar form of a complex number?
The polar form writes z as r(cos θ + i sin θ), where r is the modulus and θ is the argument. The exponential form r·e^(iθ) is the same thing written using Euler's formula.
Why are there two square roots?
Just as 4 has square roots 2 and −2, every non-zero complex number has two square roots that are negatives of each other. The calculator shows both as w₁ and w₂.
What is the conjugate used for?
The conjugate of a + bi is a − bi. Multiplying a number by its conjugate gives the real number a² + b², which is why conjugates are used to divide complex numbers and to find reciprocals.
